ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A paired contrast with unequal unit-level effects

Example

Suppose $Y_{1} (0) = Y_{2} (0) = 0$ , $Y_{1} (1) = 4$ , and $Y_{2} (1) = 2$ . The unit effects are $δ_{1} = 4$ and $δ_{2} = 2$ . Under the two equally likely assignments, the treated-minus-control contrasts are $4$ and $2$ . Their mean is the pair average effect $3$ and their design variance is $1$ . The centered paired-contrast values are $+ 1$ and $- 1$ with equal weights, so their variance is $1$ .

Verification

Given: A two-unit, two-treatment pair with no interference, $Y_{1} (0) = Y_{2} (0) = 0$ , $Y_{1} (1) = 4$ , and $Y_{2} (1) = 2$ .

[F1] In the $t = 2, b = 1$ RCBD, the two treatment permutations are equally likely (Randomized complete-block design).

[F2] On a finite probability space, expectation is the weighted sum of the values (Expectation of a real random variable on a finite probability space).

[F3] Variance is the expectation of the squared centered value (Moments, variance, and covariance on a probability space).

1.1givenF1

If unit $1$ receives treatment, the contrast is $Y_{1} (1) - Y_{2} (0) = 4$ ; if unit $2$ receives treatment, it is $Y_{2} (1) - Y_{1} (0) = 2$ . They are equally likely by [F1].

2.1F1F2F3step 1.1algebra∎

By [F2] the contrast mean is $(4 + 2) / 2 = 3$ , the pair average effect. By [F3] its variance is ${(4 - 3)^{2} + (2 - 3)^{2}} / 2 = 1$ . Thus the example has unequal unit effects and directly computes the observed contrast moments.

Depends on

Used by

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Sources

Susan Holmes and Wolfgang Huber, Modern Statistics for Modern Biology, Chapter 13 §13.4.4 (standard reference, not scraped)